Clinical statistics for non-statisticians: Day one

Introduction

Tell us one interesting number about yourself
Examples
- I have traveled to eight countries outside the United States
  - (Canada, Italy, China, France, Russia, England, Holland, and Iceland)
- I did not learn how to drive until I was 29 years old
- My highest chess rating was 1802, but I am not that good any more.

Your turn

A bit more about myself

PhD in Statistics in 1982 from the University of Iowa
Currently full professor
Part-time statistical consultant
Funded on 18 research grants
Over 100 peer-reviewed publications
Website with over 2,000 pages
Many invitations to talk at conferences

I have a PhD in Statistics from the University of Iowa. I have always had a strong interest in the computational side of Statistics. My dissertation was 150 pages, and 100 of those pages were computer generated graphs.

I am currently a full professor at the University of Missouri-Kansas City in the Department of Biomedical and Health Informatics. I also do statistical consulting on a part-time basis.

I have been a prolific researcher, receiving support from 18 different grants, and writing over 100 peer-reviewed publications.

I started a website in 1998, writing about data analysis, research ethics, and evidence based medicine. I wrote about two or three pages every week and my site now has over 2,000 pages. It shows the value of persistence.

I love to talk about Statistics and have given many presentations at regional, national, and international conferences. This ranges from short 15 minute talks to day long short courses.

Outline of the three day course

Day one: Numerical summaries and data visualization
Day two: Hypothesis testing and sampling
Day three: Statistical tests to compare treatment to a control and regression models

My goal: help you to become a better consumer of statistics

Day one topics

Numerical summaries
- When should you present the mean versus the median
- When should you present the range versus standard deviation
- How should you display percentages
- Why should you round liberally

Day one topics (continued)

Data visualization
- How should you display continuous data
- Why is the normal bell-shaped curve important
- How should you display categorical data
- How do you illustrate trends and patterns
- What are some common mistakes in the choice of colors

Counting and proportions

Counts are the most common statistic
- Counts are error prone
- Counts require a solid operational definition

Student exercise

Count the number of occurrences of the letter “e”.

A quality control  program is easiest
to implement from the top down. 
Make sure that you understand the 
the commitment of time and money
that is involved. Every workplace is
different, but think about allocating
10% of your time and 10% of the 
time of all your employees to 
quality control.

Counting sperm

Figure 1: Image of a haemocytometer

Speaker notes

This image is take from the WHO laboratory manual for the examination and processing of human semen, published in 2021. It shows a haemocytometer, an instrument used for counting the number of cells. To get a proper count, you need to include any cells inside the four by four grid of large squares in the middle of this micrograph. But what does “inside” mean? Should you count only those cells entirely inside the four by four grid. Or should you include cells that are partially inside the grid?

One rule is to count cells if the head of the sperm cell touches the top or right side of a square, but not if it touches the bottom or left side of the square. And don’t count a sperm cell if only the tail is inside the square.

That’s not the only way you can do this, but just make sure that whatever convention you use for deciding “inside” versus “outside” is consistent across your laboratory.

Tables of counts, using the Titanic data.

Figure 2: Counts of survival by gender

Percentages dividing by column totals

Figure 3: Column percentages

Percentages dividing by row totals

Row percentages

Percentages divided by grand total

Cell percentages

My recommendations

Treatment or exposure as rows
Outcome as columns
Usually report row percentages
- Female mortality rate: 33%
- Male mortality rate: 83%
But sometimes column percentages
- Survivors: 68% female, 32% male

Some rationale for these choices

My way

               Survived
               No          Yes
Sex Female     33% (154)   67% (308)
    Male       83% (863)   17% (142)

Not my way

               Sex
               Female      Male
Survived  No   33% (154)   83% (863)
          Yes  67% (308)   17% (142)

On your own

Calculate row and column percentages for the following tables. Interpret your results.

Figure 4: Titanic passenger class counts

The mean (average)

Figure 6: Cartoon image of Professor Mean

The median

Figure 7: Road with a median strip

Calculation of the mean and median

Mean
- Add up all the values, divide by the sample size
Median
- Sort the data
  - Select the middle value if n is odd
  - go halfway between the two middle values if n is even

Formal mathematical definitions

Mean
- \(\bar{X}=\frac{1}{n}\Sigma X_i\)
Median
- Sorted values \(X_{[1]},X_{[2]},...,X_{[n]}\)
  - \(X_{[(n+1)/2]}\) if n is odd,
  - \((X_{[n/2]}+X_{[n/2+1]})/2\) if n is even

Bacteria before and after A/C upgrade

Room Before  After Change
 121   11.8   10.1   -1.7
 125    7.1    3.8   -3.3
 163    8.2    7.2   -1.0
 218   10.1   10.5    0.4
 233   10.8    8.3   -2.5
 264   14     12     -2.0
 324   14.6   12.1   -2.5
 325   14     13.7   -0.3

Calculation of the mean and median

Mean
- Add up all the values
- Divide by the sample size
Median
- Sort the data

Before remediation mean

11.8 + 7.1 + 8.2 + 10.1 + 10.8 + 14 + 14.6 + 14 = 90.6

90.6 / 8 = 11.325

Round to 11.3

After remediation mean

10.1 + 3.8 + 7.2 + 10.5 + 8.3 + 12 + 12.1 + 13.7 = 77.7

77.7 / 8 = 9.7125

Round to 9.7

Before remediation median (1/4)

Before remediation median (2/4)

Before remediation median (3/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  10.8
  
121  11.8  11.8
  
264  14.0  
  
325  14.0  
  
324  14.6

Before remediation median (4/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  10.8
                  (10.8 + 11.8) / 2 = 11.3
121  11.8  11.8
  
264  14.0  
  
325  14.0  
  
324  14.6

After remediation median (1/4)

After remediation median (2/4)

After remediation median (3/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  10.1
  
218  10.5  10.5
  
264  12.0  
  
324  12.1  
  
325  13.7

After remediation median (4/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  10.1
                  (10.1 + 10.5) / 2 = 10.3
218  10.5  10.5
  
264  12.0  
  
324  12.1  
  
325  13.7

Choosing between the mean and median

When do you use the mean?
- When totals are important
When do you use the median
- When outliers/skewness might distort your conclusions
Often, either is fine

Criticisms of the mean and median

Are you combining apples and onions?
Are you ignoring minorities?

There’s a wonderful cartoon by Dana Fradon that appeared in The New Yorker in 1976. She shows a road going into town and the sign by the side of the road reads “Hillsdale, Founded 1802, Altitude 600, Population 3,700. Total 6,122.” You can’t add these things together.

It’s similar for means. There was a dataset showing housing prices for homes in Boston and none of the analyses seemed to make sense. The problem in Boston is that a small number of the houses had prices that were out of sync with their other homes. These were historical houses, such as Paul Revere’s house.

When you are averaging numbers, maybe it’s okay to have a few oranges in with the apples. A mix of apples and oranges is just fruit salad. You shouldn’t have a problem with that.

When it becomes a problem is when the data are so diverse that it becomes a mix of apples and onions. There are lots of great recipes that mix apples and oranges, but none that mix apples and onions.

The other problem is that an average may be a reasonable number to represent the majority of patients in your sample, but it may masks some important trends that appear in a minority.

This is a big problem in a larger context than just the mean or median. There are some very fancy high tech predition models that work very well for most people and the statistics like the mean and median back this up quite nicely. But the prediction models perform terribly for minority groups.

Use of the mean for ordinal data

Gould 1985

Figure 8: Gould 1985

Bridge 2001, PMID: 11405531

Figure 9: Bridge and McKenzie 2001

Bridge 2001, PMID: 11405531 (continued)

The measurement of airway resistance by the interrupter technique (Rint) needs standardization. Should measurements be made be during the expiratory or inspiratory phase of tidal breathing? In reported studies, the measurement of Rint has been calculated as the median or mean of a small number of values, is there an important difference?

Bridge 2001, PMID: 11405531 (continued)

In the present data the mean of a set of values contributing to a measurement was not significantly different from the median. However, the use of the median has been recommended since it is less affected by possible outlying values such as might be included by fully automated equipment.

Chen 2019, PMID: 31806195

Figure 10: Chen et al 2019

Chen 2019, PMID: 31806195 (continued)

Background: The prices of newly approved cancer drugs have risen over the past decades. A key policy question is whether the clinical gains offered by these drugs in treating specific cancer indications justify the price increases.

Chen 2019, PMID: 31806195 (continued)

Results: We found that between 1995 and 2012, price increases outstripped median survival gains, a finding consistent with previous literature. Nevertheless, price per mean life-year gained increased at a considerably slower rate, suggesting that new drugs have been more effective in achieving longer-term survival. Between 2013 and 2017, price increases reflected equally large gains in median and mean survival, resulting in a flat profile for benefit-adjusted launch prices in recent years.

Percentiles

Figure 11: Illustration of the 75th percentile

Some terminology

Percentile: goes from 0% to 100%
Quantile: goes from 0.0 to 1.0
- 25th percentile = 0.25 quantile
Quartiles: 25th, 50th, and 75th percentile
Lower quartile: 25th percentile
Upper quartile: 75th percentile

Before remediation upper quartile (1/4)

Before remediation upper quartile (2/4)

Before remediation upper quartile (3/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  
  
121  11.8  
  
264  14.0  14
  
325  14.0  14
  
324  14.6

Before remediation upper quartile (4/4)

125   7.1  
  
163   8.2  
  
218  10.1  
  
233  10.8  
  
121  11.8  
  
264  14.0  14
                  (14 + 14) / 2 = 14
325  14.0  14
  
324  14.6

After remediation upper quartile (1/4)

After remediation upper quartile (2/4)

After remediation upper quartile (3/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  
  
218  10.5  
  
264  12.0  12
  
324  12.1  12.1
  
325  13.7

After remediation upper quartile (4/4)

125   3.8  
  
163   7.2  
  
233   8.3  
  
121  10.1  
  
218  10.5  
  
264  12.0  12
                  (12 + 12.1) / 2 = 12.05
324  12.1  12.1
  
325  13.7

Gould 2006

Figure 12: Gould 2006

Best 2012

Figure 13: Best 2012

O’Neil 2016

Figure 14: O’Neil 2016

How companies learn your secrets

https://www.nytimes.com/2012/02/19/magazine/shopping-habits.html

Machine Bias

https://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing

Bias in Algorithms

https://fra.europa.eu/sites/default/files/fra_uploads/fra-2022-bias-in-algorithms_en.pdf